with \(q\)  summed over \(\lbrace 01, 11\rbrace \)  for the CZ gate and \(\lbrace 001, 011, 111\rbrace \)  for the C\(_2\) Z gate. Note that contrary to many other applications of the PMP to quantum optimal control [1]}, [2]}, [3]}, [4]}, [5]}, in our case the Hamiltonian depends on the control \(\varphi \)  in a non-linear way. Therefore, the optimal control is not given by a so-called “bang-bang” pulse, but by a continuous function \(\varphi (t)\) .
